Map Profiles Are Morbid Hallucinations

On a vertically exaggerated elevation profile, it is notoriously easy to exaggerate the perceived difficulty of a mountain trail. Vertical exaggeration can compress an innocent set of connected points into a lugubrious series of evil towers. The problem arises from our need to see a vast, bumpy chunk of the world represented by a tiny, smooth graph on paper. Unfortunately we tend to lose something in the translation from reality to hand-sized.

Map elevation profiles typically measure distance in miles or kilometers along a horizontal axis, and plot the corresponding elevation gains and losses in feet or meters from a vertical axis at specific points. Assuming an ample set of vertical points, the resulting picture emulates the summits and gaps along the trail's distance. However, keep in mind, an elevation profile only has limited inches of space to represent dozens of miles of distance horizontally and thousands of feet of elevation vertically. It is more art than science.

The design of an elevation profile's horizontal distance scale is fairly straight forward. An inch is one mile or ten kilometers or some other appropriate scale unit. On the other hand or axis, the creation of the vertical elevation scale is ripe fodder for morbid hallucinations. If one inch is one mile horizontally and we keep the same scale for elevation, then 528 feet of gain over one mile is .1, or 1/10th, of an inch vertically. Such a depiction is difficult to see, and creates an opportunity for an imaginative mind. For example, our humble tenth of an inch representation pumped up five times stretches the 1/10th inch to 1/2 inch making the 528 feet look like 2,640 feet, or a half mile rise over a mile run. The creative result is easy to see, but grossly distorted. While a picture may be worth a thousand words, a vertically exaggerated profile earns its weight in speechlessness.

After we vow our abstinence from vertically exaggerated elevation profiles, we still do have a couple of ways to scan the vague unknown of a mountain trail's ups and downs. One measurement is the percent gradient of a slope. The other measures the degrees of a slope. Either measurement will give us consistency and spare us from exaggerations. Although a slope's gradient and the degrees of a slope are related, they are not the same.

Gradient is expressed as a percentage. It tells us how the average elevation changes over every hundred feet of our total distance. For example, a 10% gradient means that we have an average elevation increase of 10 feet for every 100 feet. Assume we hike 400 feet and our elevation increases 40 feet. That's a 10% slope gradient. Here's the calculation:

40 ft. / 400 ft. * 100 is 10%

slope gradient is the elevation change divded by the distance multiplied by 100

The degrees of a slope measure the upward or downward angle of the slope. Using a bit of trigonometry, we can calculate a slope's angle in degrees of the elevation change over a distance. We divide the elevation change by the distance, and then find the arctangent of the result. We can use calculators to get the arctangent or Google: 'arctangent of (40/400) in degrees'. The degrees of slope of the 40 feet increase in elevation over the 400 feet distance is:

arctangent (40/400) in degrees is 5.7 degrees

We see from our calculations that the slope gradient, 10%, is a different measurement than the 5.7 degrees of the slope. However, there is a relationship between the slope gradient and degrees of slope. For example, if we increased our elevation gain to 80 feet over 400 feet, then the gradient of the slope would be 20% and the slope's angle would be about 11.3 degrees. We doubled our elevation increase which doubled our gradient and likewise doubled our degrees of slope. Likewise tripling, a gradient of 30% is a slope of 16.7 degrees. A good way to grasp their relationship is to think about walking up stairs with the same height and width. From the bottom to the top of the stairs, we average a 100% gradient or a 45 degree slope ( total horizontal distance / total vertical distance = 1 times 100 = 100% gradient, or arctangent of (1) = 45 degrees).

Enjoy a vertically exaggerated elevation profile if you wish, but make sure you understand that it is merely a smooth, hand-sized illusion of an awesome, incomprehensible world that we choose to hike on. Don't let an elevation profile intimidate you with its vertical exaggeration. Those evil towers only exist in fearful minds. If you desire to learn about the challenges of a mountain trail's ups and down, then peruse the gradients or degrees of its slopes. Neither percentages nor degrees will stretch the truth.

Here's an example of a 7.26:1 vertically exaggerated profile from a 2004 Maine Appalachian Trail Club map marked up to show distance and elevation measurements (points A, B, C, D, E, and F). The artwork's profile measures from 50 to 70 degrees, or a gradient well over 100 percent to convey dramatic steepness. The real trail varies between about four to 15 degrees, or 8 to 26 percent gradients. That's a challenging hike, but not as intimidating as the picture looks. See the table for the slope gradients and degrees from points A, B, C, D, E, and F.

I stopped looking at the AT map profiles b/c I've found them to be mostly worthless.The profile you use as an example was a hike that I feared as much as anticipated. In the end it wasn't nearly as bad as the profile indicated. Challenging, but not that bad.Another example is the 1st 15 miles of the 100 mile wilderness northbound. It is shown as being flat on the profiles, but is in reality a series of 200-300 foot climbs and descents that wear away at you over the course of a day. One is much better served learning how to read a topo than relying on profiles.If in fact I had looked at the Katahdin profile as an actual slope gradient as you've suggested here it would have saved me some worry ahead of time. Thanks for a new way of looking at those wild rollercoaster profiles!

To compound the problem is the fact that the profiles seldom have standard axes, either vertically or horizontally. Looking from one profile to another can be quite deceiving.

The only solution for me is to capture and manipulate the profiles to create common axes. I paste each profile into a new layer, with the Zero marks one above the other. Then stretch or compress the images until the distances all line up, at say 10 miles. To normalize the elevations I pick an elevation range from one of the profiles and create a "standard rectangle" of say 1000 feet, and adjust the other profiles to match it. Often many of the elevation/distance labels become unreadable, but one set is really all you need.

It is time consuming, but worth the effort when you want to compare consecutive days or reference routes.

The Appalachian Long Distance Hikers Association (ALDHA) put together the 2009 Appalachian Trail Thru-Hikers' Companion and it is available from the ATC for $13.95 (or $11.15 for ATC members) . The ALDHA Online Companion is free (GA-NC-TN, VA, WV-MD-PA,NJ-NY, MA-VT, and NH-ME). For $10 annually you can join the ALDHA.

I think a well-done, modest vertical exaggeration could be truer to the psychological experience of hiking than a 1:1 profile, even if such profiles were practical to print. Most people seem to perceive slopes as steeper than they really are–my father used to tell stories about his young Scouts claiming they had hiked up a 45 degree hill, and I've certainly been at the top of ski slopes just beyond my comfort zone that appeared (at least at the time) to be vertical drops.

Looking at the maps and books I have close at hand, the ~3:1 exaggeration used in the Falcon guide to Hells Canyon *feels* pretty close to the actual experience of hiking those trails. Having said that, the profile on Earthwalk's map of the Wonderland Trail (~7:1) is easier for me to associate with the actual trail (I know the trails and the map better than Hells Canyon) which leads me to imagine the Hells Canyon trails as less strenuous than they really are.

For me, a good visual display of information will always trump a table of slope gradients or degrees. And some sort of profile has the potential to be a good adjunct to a topo map. But good visual information design is tough (that's why Edward Tufte can charge so much for seminars), and there's clearly plenty of room for improvement or re-imagining. I wonder if a logarithmic scale would have potential.

Interesting. Thinking about this a bit more, it seems like published profiles use the degree of vertical exaggeration as a more-or-less arbitrary variable, driven by the practicalities of image size, not trail realities. May be a missed opportunity–what if the degree of exaggeration was actually dependent on something relevant to hiking? I'm thinking of an image with a set of superimposed profiles with different degrees of exaggeration, where the ratio of exaggeration was a function of, say, packweight carried (might be interesting to plot two such profiles with more and less frequent periods of re-supply). Making vertical exaggeration a function of hiker conditioning or weather/trail conditions might also be instructive.